There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$

(by $\omega^{*}$ I mean set of natural number with reversed order). It seems to be a non-trivial result - for example, one can derive it from the Baumgartner-Hajnal theorem but this is, in my taste, too heavy machinery.

Do you know who iss responsible for this result? Are there any cheaper ways (than the BH-theorem) to obtain it?